CSCE 552 Fall 2012

1. CSCE 552 Fall 2012 Math, Physics and Collision Detection By Jijun Tang

2. Homework #2 • Major use cases of your system • Due on Wednesday Oct 17th, before class.

3. Use Cases • Use Case Name: Place Order • Actors: • Registered Shopper (Has an existing account, possibly with billing and shipping information) • Fulfillment System (processes orders for delivery to customers) • Billing System (bills customers for orders that have been placed) • Triggers: • The user indicates that she wants to purchase items that she has selected. • Preconditions: • User has selected the items to be purchased. • Post-conditions: • The order will be placed in the system. • The user will have a tracking ID for the order. • The user will know the estimated delivery date for the order. • Flow: • The user will indicate that she wants to order the items that have already been selected. • The system will present the billing and shipping information that the user previously stored. • The user will confirm that the existing billing and shipping information should be used for this order. • The system will present the amount that the order will cost, including applicable taxes and shipping charges. • The user will confirm that the order information is accurate. • The system will provide the user with a tracking ID for the order. • The system will submit the order to the fulfillment system for evaluation. • The fulfillment system will provide the system with an estimated delivery date. • The system will present the estimated delivery date to the user. • The user will indicate that the order should be placed. • The system will request that the billing system should charge the user for the order. • The billing system will confirm that the charge has been placed for the order. • The system will submit the order to the fulfillment system for processing. • The fulfillment system will confirm that the order is being processed. • The system will indicate to the user that the user has been charged for the order. • The system will indicate to the user that the order has been placed. • The user will exit the system.

4. 2D Example θ

5. Translation • A 3  3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed • We must add a translation component D to move the origin:

6. Homogeneous coordinates • Four-dimensional space • Combines 3  3 matrix and translation into one 4  4 matrix

7. Translation • Translation matrix • Translates the origin by the vector T

8. Scale • Scale matrix • Scales coordinate axes by a, b, and c • If a = b = c, the scale is uniform

9. Rotation (Z) • Rotation matrix • Rotates points about the z-axis through the angle q

10. Rotations (X, Y) • Similar matrices for rotations about x, y

11. Geometry -- Lines • A line in 3D space is represented by • S is a point on the line, and V is the direction along which the line runs • Any point P on the line corresponds to a value of the parameter t • Two lines are parallel if their direction vectors are parallel

12. Plane • A plane in 3D space can be defined by a normal direction N and a point P • Other points in the plane satisfy N Q P

13. Distance from Point to a Line • Distance d from a point P to a lineS + t V P d S V

14. Line and Plane Intersection • Let P(t) = S + t V be the line • Let L = (N, D) be the plane • We want to find t such that LP(t) = 0 • Careful, S has w-coordinate of 1, and V has w-coordinate of 0

15. Formular • If LV = 0, the line is parallel to the plane and no intersection occurs • Otherwise, the point of intersection is

16. Real-time Physics in Game at Runtime: • Enables the emergent behavior that provides player a richer game experience • Potential to provide full cost savings to developer/publisher • Difficult • May require significant upgrade of game engine • May require significant update of asset creation pipelines • May require special training for modelers, animators, and level designers • Licensing an existing engine may significantly increase third party middleware costs

17. Engines • Commercial • Game Dynamics SDK (Havok.com) • Renderware Physics (renderware.com) • NovodeX SDK (novodex.com) • Free • Open Dynamic Engine (ODE) (ode.org) • Tokamak Game Physics SDK (tokamakphysics.com) • Newton Game Dynamics SDK (newtondynamics.com) • Unity

18. Particle Physics • What is a Particle? • A sphere of finite radius with a perfectly smooth, frictionless surface • Experiences no rotational motion (or assume the sphere has no size) • Particle Kinematics • Defines the basic properties of particle motion • Position, Velocity, Acceleration

19. Particle Position • Location of Particle in World Space • SI Units: meters (m) • Changes over time when object moves

20. Particle Velocity and Acceleration • Velocity (SI units: m/s) • First time derivative of position: • Acceleration (SI units: m/s2) • First time derivative of velocity • Second time derivative of position

21. Newton’s 2nd Law of Motion • Paraphrased –“An object’s change in velocity is proportional to an applied force” • The Classic Equation: • m = mass (SI units: kilograms, kg) • F(t) = force (SI units: Newtons)

22. What is Physics Simulation? • The Cycle of Motion: • Force, F(t), causes acceleration • Acceleration, a(t), causes a change in velocity • Velocity, V(t) causes a change in position • Physics Simulation:Solving variations of the above equations over time to emulate the cycle of motion

23. Concrete Example: Target Practice Projectile Launch Position, pinit Target

24. Target Practice Choose Vinit to Hit a Stationary Target • ptarget is the stationary target location • We would like to choose the initial velocity, Vinit, required to hit the target at some future time, thit. • Here is our equation of motion at time thit:

25. Equation Problem • Solution in general is a bit tedious to derive… • Infinite number of solutions! • Hint: Specify the magnitude of Vinit, solve for its direction

26. Example 1 Vinit = 25 m/s Value of Radicand of tanf equation: Launch angle f: 19.4 deg or 70.6 deg 969.31

27. Finite Difference Methods-I • The Explicit Euler Integrator: • Properties of object are stored in a state vector, S • Use the above integrator equation to incrementally update S over time as game progresses • Must keep track of prior value of S in order to compute the new • For Explicit Euler, one choice of state and state derivative for particle:

28. Finite Difference Methods-II • The Verlet Integrator: • Must store state at two prior time steps, S(t) and S(t-Dt) • Uses second derivative of state instead of the first • Valid for constant time step only (as shown above) • For Verlet, choice of state and state derivative for a particle:

29. Errors Exact Euler

30. Generalized Rigid Bodies Key Differences from Particles • Not necessarily spherical in shape • Position, p, represents object’s center-of-mass location • Surface may not be perfectly smooth and friction forces may be present • Experience rotational motion in addition to translational (position only) motion

31. Additional forces • Linear Spring • Viscous Damping • Aerodynamic Drag • Friction • …

32. Linear Springs

33. Viscous Damping

34. Aerodynamic Drag S: projected front area CD: drag coefficient

35. Friction

36. Collision Detection and Resolution

37. What is Collision Detection • A fundamental problem in computer games, computer animation, physically-based modeling, geometric modeling, and robotics. • Including algorithms: • To check for collision, i.e. intersection, of two given objects • To calculate trajectories, impact times and impact points in a physical simulation.

38. Collision Detection • Complicated for two reasons • Geometry is typically very complex, potentially requiring expensive testing • Naïve solution is O(n2) time complexity, since every object can potentially collide with every other object • Two basic techniques • Overlap testing: Detects whether a collision has already occurred • Intersection testing: Predicts whether a collision will occur in the future

39. Overlap Testing (a posteriori) • Overlap testing: Detects whether a collision has already occurred, sometime is referred as a posteriori • Facts • Most common technique used in games • Exhibits more error than intersection testing • Concept • For every (small) simulation step, test every pair of objects to see if they overlap • Easy for simple volumes like spheres, harder for polygonal models

40. Overlap Testing Results • Useful results of detected collision • Pairs of objects will have collision • Time of collision to take place • Collision normal vector • Collision time calculated by moving object back in time • until right before collision • Bisection is an effective technique

41. Bisect Testing: collision detected

42. Bisect Testing: Iteration I

43. Bisect Testing: Iteration II

44. Bisect Testing: Iteration III

45. Bisect Testing: Iteration IV

46. Bisect Testing: Iteration V Time right before the collision

47. Overlap Testing: Limitations Fails with objects that move too fast • Thin glass vs. bulltes • Unlikely to catch time slice during overlap

48. Solution for This Limitation • Speed of the fastest object multiplies the time step should be smaller than the smallest objects in the scene • Possible solutions • Design constraint on speed of objects: hard to apply without affecting the play • Reduce simulation step size: too expensive

49. Intersection Testing (a priori) • Predict future collisions • When predicted: • Move simulation to time of collision • Resolve collision • Simulate remaining time step

50. Intersection Testing:Swept Geometry • Extrude geometry in direction of movement • Swept sphere turns into a “capsule” shape